This paper provides a new way of developing the fast iterative shrinkage/thresholding algorithm (FISTA) [A. Beck and M. Teboulle, SIAM T. Imaging Sci., 2 (2009), pp. 183-202] that is widely used for minimizing composite convex functions with a nonsmooth term such as the l(1) regularizer. In particular, this paper shows that FISTA corresponds to an optimized approach to accelerating the proximal gradient method with respect to a worst-case bound of the cost function. This paper then proposes a new algorithm that is derived by instead optimizing the step coefficients of the proximal gradient method with respect to a worst-case bound of the composite gradient mapping. The proof is based on the worst-case analysis called the performance estimation problem in [Y. Drori and M. Teboulle, Math. Program., 145 (2014), pp. 451-482].